The combined use of a nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems

The approximation of two-phase Stefan problems in 2-D by a nonlinear Chernoff formula combined with a regularization procedure is analyzed. The first technique allows the associated strongly nonlinear parabolic P. D. E. to be approximated by a sequence of linear elliptic problems. In addition, non-degeneracy properties can be properly exploited through the use of a smoothing process. A fully discrete scheme involving piecewise linear and constant finite elements is proposed. Energy error estimates are proven for both physical variables, namely enthalpy and temperature. These rates of convergence improve previous results.